The discussion centers on the cardinality of infinite sets and their subsets. It establishes that if a set has cardinality m, then no subset can have a cardinality greater than m, even in the case of infinite sets. This conclusion is supported by the Cantor–Bernstein–Schroeder theorem, which asserts that an injection from a set A to a subset B cannot exist if B has a greater cardinality than A. The conversation emphasizes the importance of defining cardinality m in specific contexts.
PREREQUISITESMathematicians, students of set theory, and anyone interested in the properties of infinite sets and cardinality.
Of course. If a subset $B$ of $A$ has cardinality strictly greater than the cardinality of $A$ itself, then there is an injection from $A$ to $B$, but not from $B$ to $A$, by the Cantor–Bernstein–Schroeder theorem. For an infinite setm, it is possible to have an injection into a proper subset, but there is also a trivial injection (inclusion) from a subset to the whole set.lamsung said:If a set has cardinality m then none of its subsets has cardinality greater than m.
Is it necessarily true for a infinite set case?